Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 73.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 73.0 when he fully retires, he will begin to make annual withdrawals of $192,765.00 from his retirement account until he turns 95.00. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 10.00% interest rate.
To calculate the required contributions Derek needs to make to his retirement account in order to achieve his retirement goal, we need to break down the problem into several steps. Derek’s retirement plan involves both contributions during his working years and withdrawals during his retirement years. We’ll compute these contributions step by step.
Determine the Retirement Account’s Future Value
First, we need to calculate the future value of Derek’s retirement account when he starts making withdrawals at age 73. This calculation takes into account the interest rate of 10.00% per year. The formula for calculating the future value of a series of equal payments is:
��=���∗[(1+�)�−1]/�
Where:
�� = Future Value of the retirement account at age 73.
��� = Annual withdrawal amount after retirement ($192,765.00).
� = Annual interest rate (10.00% or 0.10).
� = Number of years Derek will make withdrawals (from age 73 to 95, which is 23 years).
Plugging in the values:
FV = $192,765.00 * [(1 + 0.10)^{23} – 1] / 0.10
Calculate this to find the future value of Derek’s retirement account at age 73.
Calculate the Present Value of Future Withdrawals
Now that we know the future value of Derek’s retirement account at age 73, we need to calculate the present value of his future withdrawals back to age 73. The present value formula is:
��=��/(1+�)�
Where:
�� = Present Value of future withdrawals (at age 73).
�� = Future Value calculated in step 1.
� = Annual interest rate (10.00% or 0.10).
� = Number of years between Derek’s 73rd and 95th birthdays (23 years).
Calculate this to find the present value of Derek’s future withdrawals.
Determine Contributions Needed During Working Years
Derek plans to retire at age 65. He will make contributions from his 26th birthday to his 65th birthday, which is a span of 40 years. We will use the formula for calculating the required periodic contributions to accumulate a certain future value:
���=��/[(1−(1+�)−�)/�]
Where:
��� = Required annual contribution during working years.
�� = Present Value of future withdrawals (calculated in step 2).
� = Annual interest rate (10.00% or 0.10).
� = Number of years Derek will make contributions (40 years).
Calculate this to find the annual contributions Derek needs to make during his working years to meet his retirement goal.
Final Calculation:
Once you’ve calculated the annual contributions needed during working years, you will have the amount Derek must contribute annually from his 26th birthday to his 65th birthday to reach his retirement goal.
In summary, Derek needs to calculate the future value of his retirement account, determine the present value of future withdrawals, and then compute the required annual contributions during his working years to ensure he can sustain his retirement plan. Given the specified interest rate and other parameters, this calculation will provide the exact annual contributions Derek needs to make to achieve his retirement goal.
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